×

Graded extensions of monoidal categories. (English) Zbl 0988.18004

In this note the authors generalize the well known results of Schreier-Eilenberg-Mac Lane on group extensions to a categorical level. More precisely, let \(\Gamma \) be a group and \(({\mathcal C}, \otimes)\) a monoidal category. The authors show that any \(\Gamma \)-monoidal extension of \(({\mathcal C}, \otimes)\) is equivalent to a crossed product extension \((\Delta (\theta , F), j)\) for some factor set \((\theta , F)\). Moreover, the map that carries a factor set \((\theta , F)\) to the corresponding crossed product extension \(\Delta (\theta , F)\) provides a bijection from the non-abelian cohomology set of \(\Gamma \) over \(({\mathcal C}, \otimes)\), \(\mathbb{H}(\Gamma , ({\mathcal C}, \otimes))\), to the set of equivalence classes of \(\Gamma \)-monoidal extensions of the monoidal category \(({\mathcal C}, \otimes)\), \(\text{Ext}(\Gamma , ({\mathcal C}, \otimes))\).
Secondly, they construct the Teichmüller obstruction map from the set of collective characters \(\wp \) to a certain \(3\)-dimensional group cohomology class \(\text{H}^3_{\wp }(\Gamma , Z({\mathcal C}, \otimes)^*)\) and show that the set \(\text{Ext}_{\wp }(\Gamma , ({\mathcal C}, \otimes))\) is non-empty if and only if its obstruction vanishes, and if \(\wp \) is unobstructed then \(\text{Ext}_{\wp }(\Gamma , ({\mathcal C}, \otimes))\) is a principal homogeneous space under the abelian group \(\text{H}^2_{\wp }(\Gamma , Z({\mathcal C}, \otimes)^*)\). These general results are applied to grade extensions of the monoidal category of modules over a \(k\)-bialgebra; in particular, the strongly graded bialgebras are classified by group cohomology.

MSC:

18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
20J05 Homological methods in group theory
18G15 Ext and Tor, generalizations, Künneth formula (category-theoretic aspects)
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Bass, H., Algebraic \(K\)-theory (1968), Benjamin: Benjamin Elmsford · Zbl 0174.30302
[2] Cegarra, A. M.; Garzón, A. R.; Grandjean, A. R., Graded extensions of categories, J. Pure Appl. Algebra, 154, 117-141 (2000) · Zbl 0965.18006
[3] Dade, E. C., Compounding Clifford’s theory, Ann. of Math. (2), 91, 236-290 (1970) · Zbl 0224.20037
[4] Dade, E. C., Group graded rings and modules, Math. Z., 174, 241-262 (1980) · Zbl 0424.16001
[5] J. Duskin, Non-abelian cohomology in a topos, mimeographed notes, 1989.; J. Duskin, Non-abelian cohomology in a topos, mimeographed notes, 1989.
[6] Eilenberg, S.; Mac Lane, S., Cohomology theory in abstract groups II, Ann. of Math., 48, 326-341 (1946) · Zbl 0029.34101
[7] Faith, C., Algebra: Rings, Modules and Categories I (1973), Springer-Verlag: Springer-Verlag Berlin · Zbl 0266.16001
[8] Fröhlich, A.; Wall, C. T.C., Graded monoidal categories, Compositio Math., 28, 229-285 (1974) · Zbl 0327.18007
[9] Grothendieck, A., Catégories Fibrées et Déscente, (SGAI) Expos VI. Catégories Fibrées et Déscente, (SGAI) Expos VI, Lecture Notes in Mathematics, 224 (1971), Springer-Verlag: Springer-Verlag Berlin, p. 145-194
[10] Hacque, M., Produits croiss mixtes: Extensions des groupes et extensions d’anneaux, Comm. Algebra, 19, 3933-3997 (1991)
[11] Joyal, A.; Street, R., An Introduction to Tannaka duality and quantum groups. An Introduction to Tannaka duality and quantum groups, Lecture Notes in Mathematics, 1488 (1991), Springer-Verlag: Springer-Verlag Berlin, p. 413-492 · Zbl 0745.57001
[12] Mac Lane, S., Categories for the Working Mathematician. Categories for the Working Mathematician, Graduate Texts in Mathematics, 5 (1971), Springer-Verlag: Springer-Verlag Berlin · Zbl 0232.18001
[13] Majid, S., Foundations of Quantum Group Theory (1995), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0857.17009
[14] Nastasescu, C.; Van Oystaeyen, F., On strongly graded rings and crossed products, Comm. Algebra, 10, 2085-2106 (1982) · Zbl 0502.16001
[15] Pareigis, B., Morita equivalence of module categories with tensor products, Comm. Algebra, 9, 139-157 (1981)
[16] Saavedra, N., Catégories Tannakiennes. Catégories Tannakiennes, Lecture Notes in Mathematics, 265 (1972), Springer-Verlag: Springer-Verlag Berlin · Zbl 0241.14008
[17] Schreier, O., Über die Erweiterung von Gruppen I, Monatsh. Math. Phys., 34, 165-180 (1926) · JFM 52.0113.04
[18] Sweedler, M. E., Hopf Algebras (1969), Benjamin: Benjamin Elmsford
[19] Teichmüller, O., Über die sogenannte nichtkommutative Galoissche Theorie und die Relation \(ξ_{λ,μ,ν}ξ_{λ,μν,π}ξ^λ_{μ,ν,π}=ξ_{λ,μ,νπ}ξ_{λμ,ν,π}\), Deutsche Math., 5, 138-149 (1940) · JFM 66.1208.04
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.